{"title":"科恩扩展中的滤门格尔实数集","authors":"Hang Zhang, Shuguo Zhang","doi":"10.1002/malq.202300008","DOIUrl":null,"url":null,"abstract":"<p>We prove that for every ultrafilter <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$\\mathcal {U}$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <mi>ω</mi>\n <annotation>$\\omega$</annotation>\n </semantics></math> there exists a filter <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <msup>\n <mn>2</mn>\n <mrow>\n <mo><</mo>\n <mi>ω</mi>\n </mrow>\n </msup>\n <annotation>$2^{&lt;\\omega }$</annotation>\n </semantics></math> which is <span></span><math>\n <semantics>\n <mi>U</mi>\n <annotation>$\\mathcal {U}$</annotation>\n </semantics></math>-Menger and <span></span><math>\n <semantics>\n <mrow>\n <mi>χ</mi>\n <mo>(</mo>\n <mi>F</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>b</mi>\n <mo>(</mo>\n <mi>U</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\chi (\\mathcal {F})=\\mathfrak {b}(\\mathcal {U})$</annotation>\n </semantics></math>. We show that in the Cohen model there exists such <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$\\mathfrak {d}$</annotation>\n </semantics></math> that is not Hurewicz in the <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-Cohen model where <span></span><math>\n <semantics>\n <mrow>\n <mi>κ</mi>\n <mo>></mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>$\\kappa &gt;\\omega _{1}$</annotation>\n </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n <mo><</mo>\n <mi>d</mi>\n </mrow>\n <annotation>$\\mathfrak {b}&lt;\\mathfrak {d}$</annotation>\n </semantics></math>. We also study the filter <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math> generated by the set of mutually Cohen reals in the <span></span><math>\n <semantics>\n <mi>κ</mi>\n <annotation>$\\kappa$</annotation>\n </semantics></math>-Cohen model. We prove that <span></span><math>\n <semantics>\n <mrow>\n <mi>b</mi>\n <mrow>\n <mo>(</mo>\n <mi>F</mi>\n <mo>)</mo>\n </mrow>\n <mo>=</mo>\n <msub>\n <mi>ω</mi>\n <mn>1</mn>\n </msub>\n </mrow>\n <annotation>$\\mathfrak {b}(\\mathcal {F})=\\omega _{1}$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>(</mo>\n <mi>F</mi>\n <mo>)</mo>\n <mo>=</mo>\n <mi>κ</mi>\n </mrow>\n <annotation>$\\mathfrak {d}(\\mathcal {F})=\\kappa$</annotation>\n </semantics></math> and every <span></span><math>\n <semantics>\n <msup>\n <mo>≤</mo>\n <mo>∗</mo>\n </msup>\n <annotation>$\\mathord {\\le ^{*}}$</annotation>\n </semantics></math>-dominating family in the ground model is <span></span><math>\n <semantics>\n <mi>F</mi>\n <annotation>$\\mathcal {F}$</annotation>\n </semantics></math>-unbounded in extension. Two questions are posed.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Filter-Menger set of reals in Cohen extensions\",\"authors\":\"Hang Zhang, Shuguo Zhang\",\"doi\":\"10.1002/malq.202300008\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We prove that for every ultrafilter <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$\\\\mathcal {U}$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <mi>ω</mi>\\n <annotation>$\\\\omega$</annotation>\\n </semantics></math> there exists a filter <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <msup>\\n <mn>2</mn>\\n <mrow>\\n <mo><</mo>\\n <mi>ω</mi>\\n </mrow>\\n </msup>\\n <annotation>$2^{&lt;\\\\omega }$</annotation>\\n </semantics></math> which is <span></span><math>\\n <semantics>\\n <mi>U</mi>\\n <annotation>$\\\\mathcal {U}$</annotation>\\n </semantics></math>-Menger and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>χ</mi>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>b</mi>\\n <mo>(</mo>\\n <mi>U</mi>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$\\\\chi (\\\\mathcal {F})=\\\\mathfrak {b}(\\\\mathcal {U})$</annotation>\\n </semantics></math>. We show that in the Cohen model there exists such <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math> which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character <span></span><math>\\n <semantics>\\n <mi>d</mi>\\n <annotation>$\\\\mathfrak {d}$</annotation>\\n </semantics></math> that is not Hurewicz in the <span></span><math>\\n <semantics>\\n <mi>κ</mi>\\n <annotation>$\\\\kappa$</annotation>\\n </semantics></math>-Cohen model where <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>κ</mi>\\n <mo>></mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\kappa &gt;\\\\omega _{1}$</annotation>\\n </semantics></math> is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>b</mi>\\n <mo><</mo>\\n <mi>d</mi>\\n </mrow>\\n <annotation>$\\\\mathfrak {b}&lt;\\\\mathfrak {d}$</annotation>\\n </semantics></math>. We also study the filter <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math> generated by the set of mutually Cohen reals in the <span></span><math>\\n <semantics>\\n <mi>κ</mi>\\n <annotation>$\\\\kappa$</annotation>\\n </semantics></math>-Cohen model. We prove that <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>b</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>=</mo>\\n <msub>\\n <mi>ω</mi>\\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation>$\\\\mathfrak {b}(\\\\mathcal {F})=\\\\omega _{1}$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>(</mo>\\n <mi>F</mi>\\n <mo>)</mo>\\n <mo>=</mo>\\n <mi>κ</mi>\\n </mrow>\\n <annotation>$\\\\mathfrak {d}(\\\\mathcal {F})=\\\\kappa$</annotation>\\n </semantics></math> and every <span></span><math>\\n <semantics>\\n <msup>\\n <mo>≤</mo>\\n <mo>∗</mo>\\n </msup>\\n <annotation>$\\\\mathord {\\\\le ^{*}}$</annotation>\\n </semantics></math>-dominating family in the ground model is <span></span><math>\\n <semantics>\\n <mi>F</mi>\\n <annotation>$\\\\mathcal {F}$</annotation>\\n </semantics></math>-unbounded in extension. Two questions are posed.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300008\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202300008","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove that for every ultrafilter on there exists a filter on which is -Menger and . We show that in the Cohen model there exists such which are tall by using a construction of Nyikos's [10]. These answer a question of Das [2, Problem 7]. We prove that there is a Menger filter of character that is not Hurewicz in the -Cohen model where is uncountable regular. This shows that the positive answer to a question of Hernández-Gutiérrez and Szeptycki [3, Question 2.8] is consistent with . We also study the filter generated by the set of mutually Cohen reals in the -Cohen model. We prove that and and every -dominating family in the ground model is -unbounded in extension. Two questions are posed.